Classes of Rational Graphs Christophe Morvan Irisa Journ ees - - PowerPoint PPT Presentation

Classes of Rational Graphs

Classes of Rational Graphs

Christophe Morvan

Irisa

Journ´ ees Montoises 2006

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Classes of Rational Graphs

Plan

1 Definitions

Notations Rational Graphs

2 Structure defined families

Deterministic Rational Graphs Rational Trees Rational DAG

3 Automata defined families

Automatic Graphs Monotonous Rational Graphs Commutative Rational Graphs

4 Conclusion and future works

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Classes of Rational Graphs

Plan

1 Definitions

Notations Rational Graphs

2 Structure defined families

Deterministic Rational Graphs Rational Trees Rational DAG

3 Automata defined families

Automatic Graphs Monotonous Rational Graphs Commutative Rational Graphs

4 Conclusion and future works

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Classes of Rational Graphs

Plan

1 Definitions

Notations Rational Graphs

2 Structure defined families

Deterministic Rational Graphs Rational Trees Rational DAG

3 Automata defined families

Automatic Graphs Monotonous Rational Graphs Commutative Rational Graphs

4 Conclusion and future works

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Classes of Rational Graphs

Plan

1 Definitions

Notations Rational Graphs

2 Structure defined families

Deterministic Rational Graphs Rational Trees Rational DAG

3 Automata defined families

Automatic Graphs Monotonous Rational Graphs Commutative Rational Graphs

4 Conclusion and future works

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Classes of Rational Graphs Definitions

Plan

1 Definitions

Notations Rational Graphs

2 Structure defined families

Deterministic Rational Graphs Rational Trees Rational DAG

3 Automata defined families

Automatic Graphs Monotonous Rational Graphs Commutative Rational Graphs

4 Conclusion and future works

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Classes of Rational Graphs Definitions Notations

Graphs

Graphs A graphe G is a subset of V × A × V

• V is an arbitrary set of vertices
• A is a finite set of labels

Conventions (u, a, v) is denoted by u

a

− →v For rational graphs V := X ∗ (X some finite alphabet)

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Classes of Rational Graphs Definitions Notations

Graphs

Graphs A graphe G is a subset of V × A × V

• V is an arbitrary set of vertices
• A is a finite set of labels

Conventions (u, a, v) is denoted by u

a

− →v For rational graphs V := X ∗ (X some finite alphabet)

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Classes of Rational Graphs Definitions Notations

Graphs

Graphs A graphe G is a subset of V × A × V

• V is an arbitrary set of vertices
• A is a finite set of labels

Conventions (u, a, v) is denoted by u

a

− →v For rational graphs V := X ∗ (X some finite alphabet)

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Classes of Rational Graphs Definitions Notations

Graphs

Graphs A graphe G is a subset of V × A × V

• V is an arbitrary set of vertices
• A is a finite set of labels

Conventions (u, a, v) is denoted by u

a

− →v For rational graphs V := X ∗ (X some finite alphabet)

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Classes of Rational Graphs Definitions Notations

Graphs

Graphs A graphe G is a subset of V × A × V

• V is an arbitrary set of vertices
• A is a finite set of labels

Conventions (u, a, v) is denoted by u

a

− →v For rational graphs V := X ∗ (X some finite alphabet)

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Classes of Rational Graphs Definitions Rational Graphs

Rational graphs

Definition A rational graph is defined by a rational transducer

ε A 1 00 01 11 000 001 011 111 a a a b b b b b b c c c p q1 q2 r1 r2 r3 a b c ε/0 0/1 1/A 1/ε 1/1 0/0 0/1 0/0 1/1 0/0 1/1

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Classes of Rational Graphs Definitions Rational Graphs

Rational graphs

Definition A rational graph is defined by a rational transducer

ε A 1 00 01 11 000 001 011 111 a a a b b b b b b c c c p q1 q2 r1 r2 r3 a b c ε/0 0/1 1/A 1/ε 1/1 0/0 0/1 0/0 1/1 0/0 1/1

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Classes of Rational Graphs Definitions Rational Graphs

Rational graphs

Definition A rational graph is defined by a rational transducer

ε A 1 00 01 11 000 001 011 111 a a a b b b b b b c c c p q1 q2 r1 r2 r3 a b c ε/0 0/1 1/A 1/ε 1/1 0/0 0/1 0/0 1/1 0/0 1/1

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Classes of Rational Graphs Definitions Rational Graphs

Rational graphs

Definition A rational graph is defined by a rational transducer

ε A 1 00 01 11 000 001 011 111 a a a b b b b b b c c c p q1 q2 r1 r2 r3 a b c ε/0 0/1 1/A 1/ε 1/1 0/0 0/1 0/0 1/1 0/0 1/1

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Classes of Rational Graphs Definitions Rational Graphs

Rational graphs

Definition A rational graph is defined by a rational transducer

ε A 1 00 01 11 000 001 011 111 a a a b b b b b b c c c p q1 q2 r1 r2 r3 a b c ε/0 0/1 1/A 1/ε 1/1 0/0 0/1 0/0 1/1 0/0 1/1

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Classes of Rational Graphs Definitions Rational Graphs

Classical questions

Proposition Following properties are undecidable for rational graphs: (i) Accessibility (between two vertices); (ii) Connectedness (of the whole graph); (iii) Isomorphism (of two graphs); (iv) First order theory (of a rational graph).

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Classes of Rational Graphs Definitions Rational Graphs

Traces

Example (traces)

ε A 1 00 01 11 000 001 011 111 a a a b b b b b b c c c

Theorem (Morvan Stirling 01) The traces of rational graphs between two vertices coincide with the context sensitive languages.

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Classes of Rational Graphs Definitions Rational Graphs

Traces

Example (traces)

ε A 1 00 01 11 000 001 011 111 a a a b b b b b b c c c

The trace of this graph between ε and A is {anbncn | n ∈ N} Tr(G, ε, A) = {anbncn | n ∈ N} Theorem (Morvan Stirling 01) The traces of rational graphs between two vertices coincide with the context sensitive languages.

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Classes of Rational Graphs Definitions Rational Graphs

Traces

Example (traces)

ε A 1 00 01 11 000 001 011 111 a a a b b b b b b c c c

The trace of this graph between ε and A is {anbncn | n ∈ N} Tr(G, ε, A) = {anbncn | n ∈ N} Theorem (Morvan Stirling 01) The traces of rational graphs between two vertices coincide with the context sensitive languages.

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Classes of Rational Graphs Structure defined families

Plan

1 Definitions

Notations Rational Graphs

2 Structure defined families

Deterministic Rational Graphs Rational Trees Rational DAG

3 Automata defined families

Automatic Graphs Monotonous Rational Graphs Commutative Rational Graphs

4 Conclusion and future works

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Classes of Rational Graphs Structure defined families Deterministic Rational Graphs

Deterministic Rational Graphs

Definition A rational graph is deterministic if each vertex is the source of at most on arc for each label Proposition Determinism is decidable for rational graphs Proposition First order theory and accessibility are undecidable for deterministic rational graphs

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Classes of Rational Graphs Structure defined families Deterministic Rational Graphs

Deterministic Rational Graphs

Definition A rational graph is deterministic if each vertex is the source of at most on arc for each label Proposition Determinism is decidable for rational graphs Proposition First order theory and accessibility are undecidable for deterministic rational graphs

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Classes of Rational Graphs Structure defined families Deterministic Rational Graphs

Deterministic Rational Graphs

Definition A rational graph is deterministic if each vertex is the source of at most on arc for each label Proposition Determinism is decidable for rational graphs Proposition First order theory and accessibility are undecidable for deterministic rational graphs

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Classes of Rational Graphs Structure defined families Deterministic Rational Graphs

Traces

Proposition The traces of deterministic rational graphs form a boolean algebra

• f deterministic context-sensitive languages containing non

context-free languages. Question Are there context-free languages not contained in the traces of deterministic rational graphs?

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Classes of Rational Graphs Structure defined families Deterministic Rational Graphs

Traces

Proposition The traces of deterministic rational graphs form a boolean algebra

• f deterministic context-sensitive languages containing non

context-free languages. Question Are there context-free languages not contained in the traces of deterministic rational graphs?

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Classes of Rational Graphs Structure defined families Deterministic Rational Graphs

Traces

Proposition The traces of deterministic rational graphs form a boolean algebra

• f deterministic context-sensitive languages containing non

context-free languages. Question Are there context-free languages not contained in the traces of deterministic rational graphs?

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Classes of Rational Graphs Structure defined families Rational Trees

Rational Trees

Definition (Tree)

• at most one ancestor per vertex

decidable

• a single root

decidable

• connected

undecidable

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Classes of Rational Graphs Structure defined families Rational Trees

Rational Trees

Definition (Tree)

• at most one ancestor per vertex

decidable

• a single root

decidable

• connected

undecidable

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Classes of Rational Graphs Structure defined families Rational Trees

Rational Trees

Definition (Tree)

• at most one ancestor per vertex

decidable

• a single root

decidable

• connected

undecidable

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Classes of Rational Graphs Structure defined families Rational Trees

A simple example - 2n-tree

p r1 a A/A0 0/0 r2 b A/ε 0/0 q r3 b ε/ε 0/1 1/0 1/1 0/0

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Classes of Rational Graphs Structure defined families Rational Trees

A simple example - 2n-tree

p r1 a A/A0 0/0 r2 b A/ε 0/0 q r3 b ε/ε 0/1 1/0 1/1 0/0 A A0 A00 A000 a a a

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Classes of Rational Graphs Structure defined families Rational Trees

A simple example - 2n-tree

p r1 a A/A0 0/0 r2 b A/ε 0/0 q r3 b ε/ε 0/1 1/0 1/1 0/0 A A0 A00 A000 a a a ε 00 000 b b b b

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Classes of Rational Graphs Structure defined families Rational Trees

A simple example - 2n-tree

p r1 a A/A0 0/0 r2 b A/ε 0/0 q r3 b ε/ε 0/1 1/0 1/1 0/0 A A0 A00 A000 a a a ε 00 000 b b b b 1 01 10 11 001 b b b b b

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Classes of Rational Graphs Structure defined families Rational Trees

Properties of rational trees

Proposition For rational trees

• inclusion and equality is decidable
• accessibility (between vertices) is decidable
• effective rational set of leaves
• some sub-trees are non-rational

Theorem (Carayol Morvan 06)

• First order theory of rational trees decidable
• First order theory with accessibility of rational trees

undecidable

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Classes of Rational Graphs Structure defined families Rational Trees

Properties of rational trees

Proposition For rational trees

• inclusion and equality is decidable
• accessibility (between vertices) is decidable
• effective rational set of leaves
• some sub-trees are non-rational

Theorem (Carayol Morvan 06)

• First order theory of rational trees decidable
• First order theory with accessibility of rational trees

undecidable

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Classes of Rational Graphs Structure defined families Rational Trees

Properties of rational trees

Proposition For rational trees

• inclusion and equality is decidable
• accessibility (between vertices) is decidable
• effective rational set of leaves
• some sub-trees are non-rational

Theorem (Carayol Morvan 06)

• First order theory of rational trees decidable
• First order theory with accessibility of rational trees

undecidable

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Classes of Rational Graphs Structure defined families Rational Trees

Properties of rational trees

Proposition For rational trees

• inclusion and equality is decidable
• accessibility (between vertices) is decidable
• effective rational set of leaves
• some sub-trees are non-rational

Theorem (Carayol Morvan 06)

• First order theory of rational trees decidable
• First order theory with accessibility of rational trees

undecidable

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Classes of Rational Graphs Structure defined families Rational Trees

Properties of rational trees

Proposition For rational trees

• inclusion and equality is decidable
• accessibility (between vertices) is decidable
• effective rational set of leaves
• some sub-trees are non-rational

Theorem (Carayol Morvan 06)

• First order theory of rational trees decidable
• First order theory with accessibility of rational trees

undecidable

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Classes of Rational Graphs Structure defined families Rational Trees

Properties of rational trees

Proposition For rational trees

• inclusion and equality is decidable
• accessibility (between vertices) is decidable
• effective rational set of leaves
• some sub-trees are non-rational

Theorem (Carayol Morvan 06)

• First order theory of rational trees decidable
• First order theory with accessibility of rational trees

undecidable

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Classes of Rational Graphs Structure defined families Rational Trees

Traces

Definition The trace of a tree is the set of path labels from the root to the leaves Conjecture There are context-free languages which are not the trace of any rational tree Example The set of words having the same number of a’s and b’s seems not to be the trace of any rational tree

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Classes of Rational Graphs Structure defined families Rational Trees

Traces

Definition The trace of a tree is the set of path labels from the root to the leaves Conjecture There are context-free languages which are not the trace of any rational tree Example The set of words having the same number of a’s and b’s seems not to be the trace of any rational tree

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Classes of Rational Graphs Structure defined families Rational Trees

Traces

Definition The trace of a tree is the set of path labels from the root to the leaves Conjecture There are context-free languages which are not the trace of any rational tree Example The set of words having the same number of a’s and b’s seems not to be the trace of any rational tree

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Classes of Rational Graphs Structure defined families Rational Trees

Curious fact

Example Given any context-sensitive language L we may construct such a rational graph (green and red states form rational sets)

u ∈ L u ∈ L u ∈ L u ∈ L

Proposition Given any context-sensitive language L (with alphabet X), there is a rational tree T labelled on X ∪ {#} such that Tr(T) ∩ X ∗ = L

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Classes of Rational Graphs Structure defined families Rational Trees

Curious fact

Example Given any context-sensitive language L we may construct such a rational graph (green and red states form rational sets)

u ∈ L u ∈ L u ∈ L u ∈ L

Proposition Given any context-sensitive language L (with alphabet X), there is a rational tree T labelled on X ∪ {#} such that Tr(T) ∩ X ∗ = L

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Classes of Rational Graphs Structure defined families Rational Trees

Curious fact

Example Given any context-sensitive language L we may construct such a rational graph (green and red states form rational sets)

u ∈ L u ∈ L u ∈ L u ∈ L

Proposition Given any context-sensitive language L (with alphabet X), there is a rational tree T labelled on X ∪ {#} such that Tr(T) ∩ X ∗ = L

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Classes of Rational Graphs Structure defined families Rational Trees

Curious fact

Example Given any context-sensitive language L we may construct such a rational graph (green and red states form rational sets)

u ∈ L u ∈ L u ∈ L u ∈ L ε ε ε

Proposition Given any context-sensitive language L (with alphabet X), there is a rational tree T labelled on X ∪ {#} such that Tr(T) ∩ X ∗ = L

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Classes of Rational Graphs Structure defined families Rational Trees

Curious fact

Example Given any context-sensitive language L we may construct such a rational graph (green and red states form rational sets)

u ∈ L u ∈ L u ∈ L u ∈ L ε ε ε #

Proposition Given any context-sensitive language L (with alphabet X), there is a rational tree T labelled on X ∪ {#} such that Tr(T) ∩ X ∗ = L

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Classes of Rational Graphs Structure defined families Rational Trees

Curious fact

Example Given any context-sensitive language L we may construct such a rational graph (green and red states form rational sets)

u ∈ L u ∈ L u ∈ L u ∈ L ε ε ε #

Proposition Given any context-sensitive language L (with alphabet X), there is a rational tree T labelled on X ∪ {#} such that Tr(T) ∩ X ∗ = L

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Classes of Rational Graphs Structure defined families Rational Trees

Rational forests

Proposition It is undecidable to know if a rational graph is a forest Proposition The first order theory of rational forests is decidable Proposition The trace of rational forests between a root and leaves coincide with context-sensitive languages

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Classes of Rational Graphs Structure defined families Rational Trees

Rational forests

Proposition It is undecidable to know if a rational graph is a forest Proposition The first order theory of rational forests is decidable Proposition The trace of rational forests between a root and leaves coincide with context-sensitive languages

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Classes of Rational Graphs Structure defined families Rational Trees

Rational forests

Proposition It is undecidable to know if a rational graph is a forest Proposition The first order theory of rational forests is decidable Proposition The trace of rational forests between a root and leaves coincide with context-sensitive languages

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Classes of Rational Graphs Structure defined families Rational DAG

Rational Directed Acyclic Graphs

Definition A rational DAG is a rational graph with no cycle Proposition It is undecidable to know if a rational graph is a DAG Proposition There is a connected rational DAG with undecidable first-order theory

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Classes of Rational Graphs Structure defined families Rational DAG

Rational Directed Acyclic Graphs

Definition A rational DAG is a rational graph with no cycle Proposition It is undecidable to know if a rational graph is a DAG Proposition There is a connected rational DAG with undecidable first-order theory

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Classes of Rational Graphs Structure defined families Rational DAG

Rational Directed Acyclic Graphs

Definition A rational DAG is a rational graph with no cycle Proposition It is undecidable to know if a rational graph is a DAG Proposition There is a connected rational DAG with undecidable first-order theory

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Classes of Rational Graphs Automata defined families

Plan

1 Definitions

Notations Rational Graphs

2 Structure defined families

Deterministic Rational Graphs Rational Trees Rational DAG

3 Automata defined families

Automatic Graphs Monotonous Rational Graphs Commutative Rational Graphs

4 Conclusion and future works

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Classes of Rational Graphs Automata defined families Automatic Graphs

Automatic Graphs

Definition A graph over X ∗ × A × X ∗ is automatic if it is recognized by a letter-to-letter transducer followed by a terminal function taking values in (Rat(X ∗) × {ε}) ∪ ({ε} × Rat(X ∗)) Example The previous examples (anbncn-graph and 2n-tree) are automatic Proposition (Hodgson 83) The first order theory of automatic graphs is decidable Theorem (Rispal 01) The traces of automatic graphs are context-sensitive languages

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Classes of Rational Graphs Automata defined families Automatic Graphs

Automatic Graphs

Definition A graph over X ∗ × A × X ∗ is automatic if it is recognized by a letter-to-letter transducer followed by a terminal function taking values in (Rat(X ∗) × {ε}) ∪ ({ε} × Rat(X ∗)) Example The previous examples (anbncn-graph and 2n-tree) are automatic Proposition (Hodgson 83) The first order theory of automatic graphs is decidable Theorem (Rispal 01) The traces of automatic graphs are context-sensitive languages

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Classes of Rational Graphs Automata defined families Automatic Graphs

Automatic Graphs

Definition A graph over X ∗ × A × X ∗ is automatic if it is recognized by a letter-to-letter transducer followed by a terminal function taking values in (Rat(X ∗) × {ε}) ∪ ({ε} × Rat(X ∗)) Example The previous examples (anbncn-graph and 2n-tree) are automatic Proposition (Hodgson 83) The first order theory of automatic graphs is decidable Theorem (Rispal 01) The traces of automatic graphs are context-sensitive languages

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Classes of Rational Graphs Automata defined families Automatic Graphs

Automatic Graphs

Definition A graph over X ∗ × A × X ∗ is automatic if it is recognized by a letter-to-letter transducer followed by a terminal function taking values in (Rat(X ∗) × {ε}) ∪ ({ε} × Rat(X ∗)) Example The previous examples (anbncn-graph and 2n-tree) are automatic Proposition (Hodgson 83) The first order theory of automatic graphs is decidable Theorem (Rispal 01) The traces of automatic graphs are context-sensitive languages

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Classes of Rational Graphs Automata defined families Monotonous Rational Graphs

Monotonous Rational Graphs

Definition A rational graph is monotonous if for each transition u/v of its transducer either satisfy |u| |v| or |u| |v| Proposition Accessibility is decidable for monotonous graphs Question Is the first order theory of monotonous graphs decidable?

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Classes of Rational Graphs Automata defined families Monotonous Rational Graphs

Monotonous Rational Graphs

Definition A rational graph is monotonous if for each transition u/v of its transducer either satisfy |u| |v| or |u| |v| Proposition Accessibility is decidable for monotonous graphs Question Is the first order theory of monotonous graphs decidable?

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Classes of Rational Graphs Automata defined families Monotonous Rational Graphs

Monotonous Rational Graphs

Definition A rational graph is monotonous if for each transition u/v of its transducer either satisfy |u| |v| or |u| |v| Proposition Accessibility is decidable for monotonous graphs Question Is the first order theory of monotonous graphs decidable?

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Classes of Rational Graphs Automata defined families Monotonous Rational Graphs

Traces of monotonous graphs

Proposition From each rational graph a bisimilar monotonous graph is computable Corollary The traces of monotonous graphs are context-sensitive languages

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Classes of Rational Graphs Automata defined families Monotonous Rational Graphs

Traces of monotonous graphs

Proposition From each rational graph a bisimilar monotonous graph is computable Corollary The traces of monotonous graphs are context-sensitive languages

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Classes of Rational Graphs Automata defined families Commutative Rational Graphs

Commutative Rational Graphs

Definition (Commutative graphs) A rational graph is commutative if its transducer is defined over a commutative monoid (Nn, Zn) Example (A commutative rational graph)

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Classes of Rational Graphs Automata defined families Commutative Rational Graphs

Commutative Rational Graphs

Definition (Commutative graphs) A rational graph is commutative if its transducer is defined over a commutative monoid (Nn, Zn) Example (A commutative rational graph)

p q1 q2 a b 2/4 2/1 3/1 2/4 2/2

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Classes of Rational Graphs Automata defined families Commutative Rational Graphs

Commutative Rational Graphs

Definition (Commutative graphs) A rational graph is commutative if its transducer is defined over a commutative monoid (Nn, Zn) Example (A commutative rational graph)

p q1 q2 a b 2/4 2/1 3/1 2/4 2/2 2 4 8 16 1 3 5 7 15 a a a b b b b b b b b4

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Classes of Rational Graphs Automata defined families Commutative Rational Graphs

Commutative Rational Graphs

Definition (Commutative graphs) A rational graph is commutative if its transducer is defined over a commutative monoid (Nn, Zn) Example (A commutative rational graph)

p q1 q2 a b 2/4 2/1 3/1 2/4 2/2 2 4 8 16 1 3 5 7 15 a a a b b b b b b b b4 6 12 b a

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Classes of Rational Graphs Automata defined families Commutative Rational Graphs

Commutative Rational Graphs

Definition (Commutative graphs) A rational graph is commutative if its transducer is defined over a commutative monoid (Nn, Zn) Example (A commutative rational graph)

p q1 q2 a b 2/4 2/1 3/1 2/4 2/2 2 4 8 16 1 3 5 7 15 a a a b b b b b b b b4 6 12 b a

Tr(G, 2, 1) = n anb2n | n ∈ N

• ∈ CF({a, b}∗)

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Classes of Rational Graphs Automata defined families Commutative Rational Graphs

Some results

Theorem (Eilenberg, Sch¨ utzenberger 69) The family of rational sets in a commutative monoid form a Boolean algebra Corollary First order theory is decidable for commutative graphs Proposition Accessibility is undecidable for commutative rational graphs Proposition The families of commutative rational graphs and prefix recognizable graphs are incomparable

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Classes of Rational Graphs Automata defined families Commutative Rational Graphs

Some results

Theorem (Eilenberg, Sch¨ utzenberger 69) The family of rational sets in a commutative monoid form a Boolean algebra Corollary First order theory is decidable for commutative graphs Proposition Accessibility is undecidable for commutative rational graphs Proposition The families of commutative rational graphs and prefix recognizable graphs are incomparable

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Classes of Rational Graphs Automata defined families Commutative Rational Graphs

Some results

Theorem (Eilenberg, Sch¨ utzenberger 69) The family of rational sets in a commutative monoid form a Boolean algebra Corollary First order theory is decidable for commutative graphs Proposition Accessibility is undecidable for commutative rational graphs Proposition The families of commutative rational graphs and prefix recognizable graphs are incomparable

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Classes of Rational Graphs Automata defined families Commutative Rational Graphs

Some results

Theorem (Eilenberg, Sch¨ utzenberger 69) The family of rational sets in a commutative monoid form a Boolean algebra Corollary First order theory is decidable for commutative graphs Proposition Accessibility is undecidable for commutative rational graphs Proposition The families of commutative rational graphs and prefix recognizable graphs are incomparable

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Classes of Rational Graphs Conclusion and future works

Plan

1 Definitions

Notations Rational Graphs

2 Structure defined families

Deterministic Rational Graphs Rational Trees Rational DAG

3 Automata defined families

Automatic Graphs Monotonous Rational Graphs Commutative Rational Graphs

4 Conclusion and future works

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Classes of Rational Graphs Conclusion and future works

Conclusion

Conclusion Several families of rational graphs

• Structural

Deterministic rational graphs Rational trees Rational DAG

• Automata driven

Future work

• Some families are promising

Deterministic rational graphs, Monotonous rational graphs, Commutative rational graphs

• Exhibit a such a family with FO+ACC decidable

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Classes of Rational Graphs Conclusion and future works

Conclusion

Conclusion Several families of rational graphs

• Structural

Deterministic rational graphs Rational trees Rational DAG

• Automata driven

Automatic graphs Monotonous rational graphs Commutative rational graphs Future work

• Some families are promising

Deterministic rational graphs, Monotonous rational graphs, Commutative rational graphs

• Exhibit a such a family with FO+ACC decidable

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Classes of Rational Graphs Conclusion and future works

Conclusion

Conclusion Several families of rational graphs

• Structural

Deterministic rational graphs Rational trees Rational DAG

• Automata driven

Automatic graphs Monotonous rational graphs Commutative rational graphs Future work

• Some families are promising

Deterministic rational graphs, Monotonous rational graphs, Commutative rational graphs

• Exhibit a such a family with FO+ACC decidable

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Classes of Rational Graphs Conclusion and future works

Conclusion

Conclusion Several families of rational graphs

• Structural

Deterministic rational graphs Rational trees Rational DAG

• Automata driven

Automatic graphs Monotonous rational graphs Commutative rational graphs Future work

• Some families are promising

Deterministic rational graphs, Monotonous rational graphs, Commutative rational graphs

• Exhibit a such a family with FO+ACC decidable

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